click drag think prefinal

8/3/2019 Click Drag Think Prefinal

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Click, drag, think!Posing and Exploring Conjectures with

Dynamic Geometry SoftwareThomas Gawlick

eibniz Universitt Hannover

Germany

ABSTRACT

We point out that to fully exploit the heuristic potential of Dynamic Geometry Software (DGS) and to

ncrease the heuristic literacy of students, extant DGS teaching units have to be ameliorated in several

ways. Thus we propose a twofold conceptual framework: heuristic reconstruction and heuristic instru-mentation of problems. Its origin is rooted in the literature, its use is demonstrated by various examples

nd its value is made plausible by a case study.

OVERVIEW

DGS has become an established tool in the mathematics classroom. Though its heuristic value is often

tressed, the literature on DGS usage reveals some shortcomings in its existing use. To see what is miss-

ng, we first reconsider models of proving and problem solving by Boero and Polya and empirical re-

earch on their viability. With these theoretical tools, we can state our main idea: to further success inroblem solving, a heuristic reconstruction of tasks is suitable. To support learners, the heuristic instru-

mentation of problems has also to be considered. For heuristic reconstruction, we adapt Polyas scheme

or problem solving. For best practice in utilizing DGS, we refine Arzarellos list of dragging modalities.

Both schemes are illustrated by detailed examples from teaching material of our ongoing research on

DGS based problem solving Finally, an advanced case study illustrates how far learners heuristic abili-

es may evolve when heuristic strategies are properly instrumented.

THEORETICAL BACKGROUND

Dynamic geometry software as a heuristic tool

Dynamic geometry software (DGS) is widely recognized as a tool of visualization that may further stu-

ents progress, see Laborde & Laborde (1995). From the beginning its heuristic role was stressed: The

hanges in the solving process brought by the dynamic possibilities of Cabri come from an active and rea-

oning visualization, from what we call an interactive process between inductive and deductive reason-

ng. (Laborde & Laborde 1992, 185) Teaching material that draws on this capability was subsequently

eveloped e.g. by Elschenbroich (1997). Meanwhile, the literature contains a plethora of proposals for the

tilization of DGS even in nongeometric situations (e.g. Gawlick 2003), various of which are accompa-

ied by reports on their outcome.

However, in a critical review of the role attributed to DGS in the literature, Hlzl (2001) concludes that

hough a lot of examples are given how DGS can support the heuristic phase of problem solving, a closer

Preliminary draft copy of:Gawlick, Th.(under review): Click, drag, think! Posing and Exploring

Conjectures with Dynamic Geometry Software, in: Habre, S. (ed.):

Dynamical mathematical software and visualization in the learning of

mathematics. Hershey, PA: IGI Global

Please do not copy, cite or circulate but use the final paper!


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ok leads one to question whether the software is really used in a methodical and activating way of

nowledge acquisition. Rather, it seems that DGS is used in a solely verifying manner: i.e. the learners

eed only vary more or less prefabricated geometric configurations to confirm empirically more or less

xplicitly stated conjectures, e.g. that the altitudes in a triangle always meet in one point.

lzl (1999, 2001) reports that the long-term application of DGS in a classroom with DGS as an integral

art of the learning environment yielded the following results:

DGS possesses a considerable heuristic potential, especially in the connection of transformationgeometry.

The application of DGS should only be realized after thorough consideration: "Dynamics per se isnot a didactical advantage."

The intensive application will be most favorable where an objective instrumental requirementmeets advanced mathematical teaching experience.

The drag mode can be used as a graphic tool, as test mode, or as a search mode. The use of the drag mode as a test mode remained shaky even after two years.

lzls results clearly show the need to enhance students capability to make heuristic use of DGS: work:

s students make only a limited use of the drag mode, up to the point where students pose nontrivial con-

ctures only from static drawings, even without attempting to verify them by dragging though the drag

ode as test mode was part of Hlzls schedule!

nalogous phenomena have been observed for CAS usage e.g. by Artigue (2002): Our attention was

tracted by the slowness and windings of this instrumental genesis ... The economic strategies of use of

e TI-92 were rarely chosen. She utilizes the theoretical framework of the instrumental approach (Veril-

n & Rabardel 1995): The instrument is differentiated from the object, material or symbolic, on which it

based and for which they use the term artefact. It is a mixed entity, constituted for the one part of antefact, and for the other, of schemes which make it an instrument. For a given individual, the artefact ate outset does not have an instrumental value. It becomes an instrument through a process, called instru-

ental genesis, by the construction of personal schemes or, more generally, the appropriation of social

e-existing schemes. This instrumental genesis works in two directions. In the first direction, instrumen-

l genesis is directed towards the artefact, loading it progressively with potentialities, and eventually

ansforming it for specific uses; this is called the instrumentalisation of the artefact. In the second direc-

on, instrumental genesis is directed towards the subject, and leads to the development or appropriation

f schemes of instrumented action which progressively constitute into techniques that permit an effective

sponse to given tasks. This latter is what is properly called instrumentation. Thus we can conceptually

ummarize Hlzls findings as follows: The instrumentation of utilization schemes for DGS like Arza-llos dragging modalities requires deliberate efforts over an extended period of time.

aborde (2001) reports similar findings for teachers from a long-term project on the integration of tech-

ology in the design of geometry tasks in planning teaching units, even teachers with DGS experience

strict the role of DGS to a facilitation of conjecture posing: The most obvious contribution of Cabri is

e possibility of dynamic visualization of geometrical relations preserved by the drag mode. Teachers

ven the novice in using technology) immediately exploited this possibility by asking students to conjec-

re properties from what they could see. However, when the students were asked to justify, the teachersd not mention the possibility of using Cabri to find a reason or to elaborate a proof. (ibid., 306)

o widen our perspective towards possible stages in finding and writing up proofs where DGS usageould profitably occur, we now take a look at process models of theorem proving and problem solving.


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esearch on theorem Proving and Problem-Solving

or the teaching and learning of proof it is necessary to guide students to master the process of proof (and

ot only to deal with finished proofs). Boero (1999) thus proposed an expert model of the proof processhat comprises the following phases:

Production of a conjecture (including exploration of the problem situation, identification of regulari-

ties, identification of conditions under which such regularities take place, identification of argumentsfor the plausibility of the produced conjecture, etc.)

) Formulation of the statement according to shared textual conventions

I) Exploration of the content (and limits of validity) of the conjecture; heuristic, semantic (or even for-

mal) elaborations about the links between hypotheses and thesis; identification of appropriate argu-ments for validation, related to the reference theory, and envisaging of possible links amongst them

V) Selection and enchaining of coherent, theoretical arguments into a deductive chain, frequently under

the guidance of analogy or in appropriate, specific cases, etc.

) Organization of the enchained arguments into a proof that is acceptable according to current mathe-matical standards

I) Approaching a formal proof. (This phase may be lacking in mathematicians' theorems.)

heorem proving may be viewed as the topmost sub discipline of mathematical problem solving. Polya

945) suggests it to be performed in four steps, namely:

1. Understanding the problem

2. Devising a plan

3. Carrying out the plan

4. Looking back

olyas first step corresponds to Boeros phases I and II, step 2 to phases III and IV, step 3 to phases V

nd VI. Polya aims at guiding the problem solver through the solution process by a sequence of questions

or each step (37 in all). Heuristic elements occur mainly in the first two steps. Schoenfeld (1985,23)ives a concise account: Such strategies include exploiting analogies, introducing auxiliary elements in a

roblem or working auxiliary problems, arguing by contradiction, working forward from the data, de-

omposing and recombining, exploiting related problems, drawing figures varying the problem, andorking backward.

mpirical research on problem solving

Neither Boeros nor Polyas model has been empirically verified in the sense that its fit to students actualehavior has been assessed, but we have empirical evidence that Boeros model is more accurate in some

etails. Namely, problem solvers in our studies frequently do consider tools and techniques that might be

seful in the proof after reading the problem. This may be attributed to phase I of Boero but is prescribed

y Polya only for step 2. However, psychological research like Newell and Simon does not support the

atural occurrence of devising a plan, and the literature is inconclusive about its benefit: The results of

orehand (1967) suggest that it might be helpful while those of Hayes (1966) indicate the contrary.

n the same vein, Kilpatrick (1967, 44) asserts that whatever merits Polya's list has for teaching problem

olving, it is of limited usefulness, as it stands, for characterizing the behavior of these subjects. sub-

ects seemingly did not exhibit behavior even remotely resembling actions suggested by the heuristicuestions. For example, no subjects asked themselves aloud whether they were using all of the essential

otions of the problem.


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ork for the macroscopic analysis of problem solving protocols from out loud problem solving ses-

ons. The idea behind the framework is to identify major turning points in a solution. This is done by

arsing a protocol into macroscopic chunks called episodes and then examining the junctures between

hem.(ibid., 314) An episode is a period of time during which an individual or a problem-solving groupengaged in one large task ... or a closely related body of tasks in the service of the same goal.(ibid.,

92)

choenfeld found that the episodes fell rather naturally into one of six categories. Their list bears a closeesemblance to the surface structure of Polyas scheme, as has been explicated and visualized by Rott

2011):

choenfeld (1992, 195) summarizes his results on problem solving as follows Approximately 60% of the

rotocols [without prior training ] were of the type ..., where the students read the problem, picked a solu-

on direction (often with little analysis or rationalization), and then pursued that approach until they ranut of time. In contrast, successful solution attempts came in a variety of shapes and sizes. This result

early shows that the heuristic literacy of students deserves some improvement. Schoenfeld (1985) re-

orts on successful training by breaking down several general heuristics to concrete strategies that are

pplicable to a narrower, but more easily identifiable class of problems. Schoenfelds results could be rep-cated with more subjects and a no-treatment-control group by Rott & Gawlick (submitted).

ONGOING RESEARCH ON HEURISTIC LITERACY

owever, while Schoenfelds framework is salient to analyze the overall structure of problem solving

rocesses and thereby predict the probability of success, it cannot provide insights into their heuristic sub-

antiality. In ongoing research on the influence of heuristic strategies on problem solving, we thus want

o further address the questions

What heuristics are present in students problem solving processes?. How do they contribute to problem solving success?i. Are heuristics helpful to activate problem solvers previous knowledge?


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Methodically, we recur to the original scheme of Polya (1945, xvi f.) reproduced here with questions

umbered for reference and two insertions from Kilpatrick (1967, 45) in brackets:

NDERSTANDING THE PROBLEM

irst. You have to understandthe problem.

What is the unknown? 2. What are the data? 3. What is the condition?

Is it possible to satisfy the condition? 5. Is the condition sufficient to determine the unknown? Or is itsufficient? 6. Or redundant? Or contradictory?

Draw a figure. 8. Introduce suitable notation.

Separate the various parts of the condition. Can you write them down?

EVISING A PLAN

econd. Find the connection between the data and the unknown. You may be obliged to consider auxil-

ry problems if an immediate connection cannot be found. You should obtain eventually aplan of the

olution.

0. Have you seen it before? 11. Or have you seen the same problem in a slightly different form?

2. Do you know a related problem? 13. Do you know a theorem that could be useful?

4. Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.

ere is a problem related to yours and solved before. Could you use it? [15. Does it have a similar un-

nown?] 16. Could you use its result? 17. Could you use its method? 18. Should you introduce some aux-

iary element in order to make its use possible?

9. Could you restate the problem? 20. Could you restate it still differently? Go back to definitions.

you cannot solve the proposed problem try to solve first some related problem. 21. Could you imagine

more accessible related problem? 22. A more general problem? 23. A more special problem? 24. An

nalogous problem? 25. Could you solve a part of the problem? 26. Keep only a part of the condition,

rop the other part; how far is the unknown then determined, how can it vary? 27. Could you deriveomething useful from the data? 28. Could you think of other data appropriate to determine the unknown?

9. Could you derive something from the unknown?] Could you change 30. the unknown or 31. data or

oth if necessary, so that the new unknown and the new data are nearer to each other?

2. Did you use all the data? 33. Did you use the whole condition? 34. Have you taken into account all

sential notions involved in the problem?

ARRYING OUT THE PLAN

hird.Carry outyour plan.

5. Carrying out your plan of the solution, check each step. 36. Can you see clearly that the step is cor-

ct? 37. Can you prove that it is correct?

ooking Back

ourth.Examine the solution obtained.

8. Can you check the result? 39. Can you check the argument?

0. Can you derive the solution differently? 41. Can you see it at a glance?

2. Can you use the result, or the method, for some other problem?

he following synopsis with the heuristic dictionary of Polya (1945) shows that heuristic strategies occur

ostly in the first two phases and that with the exception ofWorking Forward(which may be left out as a

aturally occurring strategy) and Transformation all major strategies are comprised in the scheme. (Theher questions of the scheme address metacognitive issues.)

Induction


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,25., 26. Decomposing and Recombining

8. Auxiliary Elements

4.- 18., 21.-24. Related Solved Problem, Auxiliary Problem

2. Generalization

3. Specialization4. Analogy

6., (22., 23.) Variation

27. Working Forward]

8., 29. Working Backward

bserve how many questions try to take previous knowledge into account the rationale for our question

i. being that empirical evidence suggest that more often than not problem solver fail to do this properly.

he size and complexity of Polyas scheme also makes it pretty clear that one has to devise special meansor heuristic training it is not simply a matter of handing the list over to the students and let them apply

by trial and error. Also, one would expect that acquisition of heuristic literacy takes place neitheruickly nor easily and even for experienced problem solvers it is certainly not straightforward to make

ood use of Polyas scheme, as subsequent examples will illustrate. All in all, to assimilate Polyas

cheme may well be viewed as an instance of instrumental genesis in the sense explained below. This

may account for the disappointing results of Kilpatrick (1967).

n our ongoing research, we take these considerations into account as follows:

We plan to survey the heuristic development of two classes over a period of two years

We will further the acquisition of heuristic skills by a suitable DGS usage.

n the main part of this paper, we will report on

How to design teaching units to foster heuristic literacy the idea of heuristic reconstruction.

How to integrate DGS in the acquisition of heuristic skills the concept of heuristic instrumentation.

We suppose that this rationale behind our approach can be fruitful also for others in designing DGS-based

arning environment. Details on our research design can be found in Brockmann-Behnsen (2011).

. THE CONCEPT OF HEURISTIC RECONSTRUCTION

Heuristic reconstruction means: reconsider a problem and its solution so that one can pose it (and accom-any the solution process if necessary) in such a way that enables students to make heuristic decisions on

heir own as often as possible.

is a design principle for teaching units that comprises of fourguidelines:

1. Start from a problem, often given in closed form (e.g. from a textbook), perhaps with a solution.2. To prepare for educational use, pose the following key questions:

a. What (intermediate) goals shall students reach within the course of processing the prob-lem?

b. What heuristic elements may help them to make process at a certain points? (Use e.g.Polyas list)


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c. Which of them can they find on their own? For which do they need a hint? Is the hint tobe placed in the problem statement or better given when a certain state of problem proc-

essing is reached?

d. What resources do they need to that end? Which of them can be taken as granted, whichhave to be supplied extra?

3.

Use the answer to the key questions to rework and ameliorate the original task.4. Observe how students interact with the heuristically reconstructed task and draw conclusions for

the optimization of your task format.

ote that these four guidelines bear a similarity to the four phases of Polyas scheme. And just as oneay assume that good problem solvers do pass through Polyas phases, the designers of successful teach-

ng units and textbooks will certainly have obeyed such guidelines anyway. Indeed such guidelines were

ready heeded by teaching practitioners and textbook authors with Polyas scheme in mind (Elschenbro-

h, personal communication). But since the majority of current teaching material unfolds only a small

ortion of the heuristic potential enclosed in the respective tasks it may be worthwile to state these guide-

nes explicitly.euristic reconstruction is kindred to the genetic approach: mathematics teaching ought to lead pupils

owly along the same path to higher ideas and finally to abstract formulations, as those that the human

ce in general followed from a nave primitive state to knowledge at higher, abstract level. Klein (1924,

90) The connection to heuristic becomes apparent in Wittmanns account: In order to achieve mathe-

atics teaching in which structural concepts, real mathematical problems and heuristic strategies are all

ven their full weightthe author would like to call the basic didactic conception here ' genetic' the

hoice of problems is of vital importance. Central problems will provide a starting point from which fur-

er problems and problem complexes can be generated. By these means, mathematical schemata and

euristic strategies can be learned and practiced. (Wittmann 1975, 194) The idea of empowering students

make their own decisions in the learning process is also implicit in Freudenthals approach: Good ge-metry instructionmeans leading pupils to understand why some organization, some concept, some

efinition is better than another. Traditional instruction is different All concepts, definitions and deduc-

ons are preconceived by the teacher. (Freudenthal 1973, 418)

he concept of heuristic reconstruction is elaborated best by virtue of a concrete

xampleCan any triangle be divided into two isosceles triangles?

rzarello et al. (2002, 67) states it in the following way:

ou are given a triangle ABC. Consider a point P on AB and the two triangles APC and PCB. Make a

ypothesis about the properties of ABC which are necessary so that both APC and PCB are isoscelesuch triangles are called separable).

ompare this adaption to the task format from Hlzl (1996, 175):


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What changes occur? Both authors add several heuristic elements. Firstly, they both introduce suitable

otation one of Polyas basic heuristic advices. Then, while Arzarellos students have to produce a suit-

ble drawing for themselves, Hlzl also provides two figures. Thereby he also unveils that the task isolvable at all students may take this for granted, but consider Polyas question 4. Hlzl seems to decide

at students are unlikely to pose it and provides the answer. Arzarello furthermore gives the hint that

nly some triangles are separable, by restating the task as: Find a necessary property for separable trian-

es. He may have reckoned that students analysis of the original problem will not result in such a re-atement. Furthermore his version only asks for a necessary condition the original question affords to

ttle whether it is also sufficient. Arzarello may have deemed this too difficult for the students, though

s instruction is more likely to lead to a sufficient condition by identifying a position of P where ABC

ecomes separable. Finally, Hlzls two figures suggest that there are two cases of separable triangles

is could distract students from considering the possibility of other cases, but on the other hand may be

elpful to draw their attention to the fact that there is not just one case, as in most other problems posed in

hool.

ltogether we recognize that both authors apply some sort of heuristic reconstruction, presumably accord-

g to their own unspoken guidelines. The result might be construed as mere simplification of the task

nce the task has been proven empirically to be difficult, this adaption will certainly increase the possibil-y that students can make some progress on it. But from our point of view, such changes should not be

xecuted a priori by the teacher, but explicitly and embedded in a learning situation, as to increase stu-

ents heuristic literacy.

uch a reconstruction affords to think though the original problem more thoroughly. Hefendehl-Hebeker

Hlzl (2000) provide a detailed analysis that addresses many aspects of our guidelines, including teach-

g experience. For space reasons we can give only a sketch of the heuristic reconstruction along the

bove guidelines:

Students shall recognize that some, but not all triangles are separable. Then they shall come to discover

at separable triangles fall into three (nondisjoint) classes and characterize these classes by suitable con-tions (e.g. angle conditions).

a. Students shall produce examples and perhaps counter-examples for separability. They shall identify

least the class of right-angled triangles as separable.

To come to necessary conditions, it is convenient to Work Backwards: Under what conditions can two

osceles triangles be unitedto a separable triangle? Then one can apply the heuristic Systematically

stinguish Cases: the triangles can be united along their basis or their sides respectively. From the four

onceivable cases three actually do occur, as students can discover experimentally.

To characterize the three cases by angle condition, only basic geometric knowledge is necessary. Sali-nt, however, is some proof knowledge: E.g. the class of separable triangle that consist of isosceles trian-

es united along their respective sides contains just the right-angled triangles. One needs if and only if

gumentations to show that each of the three combinatorial cases corresponds to an angle condition.

To overcome possible difficulties, students can be supplied with examples as Hlzl does, or given a

nt like Arzarellos that classes of separable triangles should be characterized by an equivalent condition.

robably it is necessary that the teacher provides the information that there are several cases and that they

annot be concisely characterized by one condition.

On the above grounds we come to choose the following approach: First, the students shall explore the

riginal situation freely for some time. Then a classroom discussion can address the following questions:Can you find other separable triangles? (Expected answer: yes. Otherwise the teacher has to give

some examples.)


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. Can you describe their properties in a way that makes it easy to decide whether a triangle belongs tothis class of separable triangles? (Expected answer: If Hlzls first case is found, it is recognized as

rectangular, but for the second case only few students will find a handy description.)

i. Does any triangle belong to these classes? (Expected answer: some students will have figured out acounter-example, where ones sees that it is not separable since it is not in the known classes of

separable triangles (i.e. either the ones found by the students or Hlzls two given classes), but with-

out having thought of how to prove that there is no other way to divide it.)

v. Are their other separable classes? (Expected answer: the students will not have thought of this possi-bility.)

How can we finally make sure that we have found all separable cases? (This may be the moment,where some students could bring up the idea to revert the line of thought: Lets assume, we have a

separable triangle. How do the isosceles triangles it consists of look like? What is the condition

[Polya 3., 30., 31.!] that they could be united? This paves the way to discover the combinatorial

cases.)

i.Now that you think that you have found all possibilities to separate a triangle, can you describe pre-cisely for the rsulting classes of triangles? (This induces the idea that the cases have different charac-

terizations, which is unlikely to be taken into consideration by all but the most experienced students.

Also, it suggests to formulate an if and only if-statement for each case.)

his classroom dialogue may well be interrupted by extended periods of single or group work, e.g. to set-

e iv. Also, teaching material may be supplied if the process demands it, e.g. the teacher may provide

orksheets that provide the formal proof structure necessary in vi. for students that do not master this

roof scheme.

he above sequencing of the teaching unit gives the students the freedom to make as many steps of the

olution process as possible on their own. Anyway, it is fundamental that the important steps in this proc-ss are made conscious and emphasized as such by the teacher, be they found by the students or intro-

uced by him. Some steps may be appreciated already during the solution process, but generally it seems

orthwhile to look back after completion of the process and to highlight the function of the decisive

eps. This is Polyas final phase experience show that students do it seldomly on their own. But doing

collectively will in the course of time give them the possibility to acquire more and more heuristic

rategies. We hypothesize that continuous efforts to make heuristics conscious will be the utmost effec-

veness factor of heuristic reconstruction.

. THE CONCEPT OF HEURISTIC INSTRUMENTATION

o far, the role of DGS in heuristic reconstruction has not been discussed. Arzarello et al. (2002) providestheoretical analysis of the role various dragging modalities can play in the solution process and exempli-

es it by the triangle separation task: Students may surmise that it is good idea to choose P as the mid-

oint of AB and construct a corresponding dynamic figure. Then they may drag C around freely to find

ases of separable triangles wandering dragging. If they try to drag C in order to realize examples oflzls first case, they perform lieu muet dragging these cases will inevitably occur when C is on the

hales circle of AB. If they mark such positions of C to make this locus visible, Arzarello speaks ofline

ragging. To validate the hypothesized connection, students may construct the circle, bind C to it and

rag it around linked dragging. (Bound dragging if the point was already constructed as semifree point

n the circle.) Arzarello assumes that an essential factor in problem solving is the interplay ofascending

nd descending control movements between the visual and the theoretical level and suggests that these areacilitated by specific dragging modalities.

GS may thus be seen as an apt means to foster heuristic strategies by suitable instrumentation of tasks.


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ays Healy et al. (1994) report degoaling phenomena: namely, the interaction with the software en-

ils a shift of attention such that students tend to leave the original task out of consideration. They sum-

marize their experience: We knew from our Logo work that freedom to create and change ones ownoal - degoaling - is double-edged. On the one hand, it allows pupils to appropriate the activities,... On the

ther hand, it sometimes enables pupils to avoid interacting with mathematics at all!

he observations of Hlzl (1996, 1999) point into the same direction: the interaction with the software

ften causes subtle differences between the tasks intended goals and the situated knowledge acquired byhe students.

herefore, not only the utilization of Polyas scheme, but also of other instruments, particularly DGS

eeds thorough consideration. We define heuristic instrumentation as the design phase of a teaching unit,

ere such considerations take place naturally on the ground of information about the learners heuristicteracy as well as their DGS proficiency. The concept is illustrated best with an

xample for combined heuristic reconstruction and instrumentation

riginal task (from Elschenbroich (1997,63)Draw anequilateral triangle. Choose an arbitrary point

side the triangle and measure the distances x,y,z to the sides. What can you say about the sum x+y+z?there any relation to other data in the triangle?

his task was criticized by Hlzl(2001,65): Such DGS activity can hardly be expected to yield more than

mpirical confirmation, nor is it likely to point to a promising solving strategy. In fact, such a strategyecessitates a mental restructuring of the triangle into appropriate equilateral triangles, thus requiring a

urposeful view that already draws on heuristic skills but hardly comes to mind by just dragging P

round. Hlzl also takes into account that Elschenbroich mentions in a footnote that one could argue

hether one had not better posed the problem in a more open manner, for example: What can you say

bout x, y, z? In doing so, however, the heuristic role of DGS would not be significantly enhanced: Drag-

ing P while watching three simultaneously changing variables does not suggest to consider their sum.ne may readily recognize the considerations of Elschenbroich as well as of Hlzl as attempts to recon-

ruct the problem and improve its instrumentation to enlarge the heuristic value of the problem. From theoint of view of our conceptual framework, we can propose the following means to that end: Accept that

he shift of focus towards x+y+z is unlikely to be performed by the students on themselves and try to gain

he heuristic value in another way:

econstructed ExampleLet ABC be an equilateral triangle and let P be on or inside ABC. Let D, E and

be feet of the perpendiculars from P to AB, AC and BC and S(P):=|DP|+|EP|+|FP|. What can you say

bout S(P)?Consider special cases!

ndeed, this restatement of the task it suitable namely for phase I and for an utilization of DGS: namely

ragging P inside or on ABC, one will readily see that S(P) is constant, as above. But to establish its pre-

se value, specialization can now be utilized: namely, by dragging P to a vertex, say C, one discovers that

n this case S(P)=|CE|=:hABC equals the length of an altitude of ABC.

eduction to special cases is also a successful strategy for phase III and IV. To that end, the helpful nota-on S(P) was introduced to the problem since it facilitates then reduction from more general P to more

pecial P. For instance, to reduce the case P on AB to the special case P=A, one may consider the triangle

BQ, where Q is the intersection point of BC with the parallel to AC through P. This observation can beompleted to a valid proof in phase IV if one considers that for the corresponding sum in PBQ we have

*(P) =PF= hPBQ and uses this to rewrite S(P) =PE+PF= d(AC, PQ) + S*(P) = hABC


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A B

C

P

Q

hPBQ

d(AC,PQ)E

F

However this line of thought is still unlikely to be discovered by students on their own. Now DGS can betilized as mediating tool by the teacher: He may advise the students to consider PBQ (or, less directively:

n equilateral triangle with vertex P) and its connection to the given problem. By dragging P along AB,

he students sees the case P=A as limit case and this may spark off the idea to reduce the case P on AB to

he special case P=A Note however, that the tool is not directly helpful to find the above representation of

(P) in terms of S*(P). This and especially the introduction of the new notation S*(P) has certainly to be

ntroduced by the teacher in a suitable way and in the right moment. This shows clearly the limits of what

an be achieved by pre-formulated instruction: a worksheet may provide a suitable starting point for a

roblem solving process, but the teacher has to monitor closely the subsequent group work and must be

repared to intervene in turning points when students are likely to get stuck.

o support teachers in handling such difficulties, Elschenbroich (2001) introduced the concept of EiWos:

lectronic interactive Worksheets. These provide the learners with pre-constructed figures to avoid the

itfalls of geometrical programming (delays by construction errors, necessity of DGS specific knowl-

dge, degoaling) and allow them to concentrate on problem solving,

Reconstructed task with EiWos support (Elschenbroich 2001)

Sum of perpendiculars

A point P is given in a regular triangle with the distances x, y, z to the sides of the tri-

ngle.) Vary P in the inside of the triangle. What do you notice about the sum of the dis-ances x + y + z?

) Parallels through P create some triangles. There you can find x und y repeatedly.

Give some reasons for the result in part a).

) Which line has the same length as x + y + z?

hint (if necessary)


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detailed account of our teaching experiences with the reconstructed version will be published else-here.

Heuristic instrumentation implicit or explicit?

n Elschenbroichs EiWos, tasks are usually broken up into a sequence of well-considered steps, with de-

ailed dragging instructions if necessary. This will entail that average teachers can guide an average learn-

ng group through the solution of a non-routine problem. The necessary heuristic strategies and their in-

rumentation, however, are all preconceived by the author so it seems questionable whether they will

e assimilated in such a way that they can be utilized in more open learning situation, where no way of

ask processing is prescribed.efore, we proposed to let the teacher support the choice of heuristics, but make the selection transparent

o the learners. In the same vein, we now propose to make the acquisition of instrumented DGS tech-

iques a conscious and deliberate one. But how to do this? The question whether dragging modalities

hould be taught has not yet been addressed in the literature. However, for best results it will be necessary

uring classroom discourse and group work as well as in instructional texts to linguistically distinguish

etween the dragging modalities in an efficient way Polyas heuristicIntroduce suitable notation.

n our ongoing research, we will contrast a learning group that sticks closely to Elschenbroichs EiWos

ith a parallel group that receives augmented instruction with respect to heuristic instrumentation. To

nvestigate these treatments we need: refined classification of dragging modalities for research and teaching

rzarello gives no empirical validation of his model and does not claim that it is all-encompassing. A

heoretical analysis suggest the following refinement: We construemodalities of dragging as consisting

fdragging acts that are performed in various articulation.

oncerning the mere dragging act, we propose a systematic classification along two dimensions:

One can distinguish, what kind of point is dragged: a base point, a semifree point (i.e. one that is bound

o a line) or a dependent point (its position is determined by the given construction, so it cannot be moved

t all). One can distinguish different ways of dragging: Namely something can be dragged away from some

bject or attempted to be moved in a certain direction or to some position. Then, one might want to move


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n object along a geometric locus be it visible or imagined. Finally, a move may not follow any appar-

nt plan at all.

his systematics is exhaustive in the following sense:

The dragged point falls necessarily into the above categories.

Any dragging will eitherpreserve a certain geometric relation, establish a new one orterminate an ex-

ting one. Such a relation may equivalently viewed as a geometric property or as a locus condition.1

hus we obtain the following taxonomy of dragging acts:

Dragging acts How is it dragged ?

What is dragged? Relation-

preserving

Relation-

terminating

Relation-

establishing

Without appar-

ent aim

Base point Through a region

or locus or to keep

a geometric prop-

erty (lieu muet)

Dragging awayfrom some object

or to abandon a

geometric property

Dragging towardsome object or to

realize a geometric

property (Guided

dragging)

Wandering drag-

ging

Semifree point Ditto Ditto Ditto Bound dragging

Dependent point Powerless dragging to check whether a point can be moved in these ways or to

demonstrate that this is impossible

ome of Arzarellos modalities fit into this scheme. These are italicized - new dragging acts are also bold-

aced.

ometimes terminating an existing relation and establishing a new one cannot be clearly distinguished orave to occur simultaneously (see example below). Thus it makes sense to combine them to relation-

hanging dragging.

ow each of these dragging acts may occur in various articulations, namely:

What is observedduring dragging? Is it the dragged point itself or is it a point or a configuration that

depends on it?

Is the observed object merely draggedor is its dragging behavior made visible in some way:

o by marking selected positions with newly constructed points,o by tracing the dragging path of the observed object,o byproducing its locus.

In what phase does the dragging act occur? We can distinguish:

o Explorative use in an ascending phaseo Verifying use in a descending phase

he salience of the refined scheme will become evident by an

For symmetry reasons one might also consider a dragging act, in which a relation continues not to be preserved.

ut this can be reduced to relation-preserving dragging if we consider instead the inverse relation. However, the


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xample 1 Consider a circle k with diameter AB and a free point C. Construct the triangle ABC. How

oes the angle at C vary with the position of C?

n exploring this situation, the problem solver would presumably start by just moving around C in some

ay or other i.e. wandering dragging (1). After some time (or by a hint) it may occur to him that inside

the angle is always acute. To check this conjecture, one would move C consciously inside k and watch

e angle Arzarellos dragging test, here construed as relation-preserving dragging act in verifying use

2). To see what happens outside C, one would move C across k relation-changing dragging in explor-ive use. (3). Moving it around there exploratively (4) would be Arzarellos guided dragging but Arza-

llo would probably dub it a dragging test if it is used to confirm the observed obtuseness. By binding C

k, the limit case can be established then bound dragging in either explorative or verifying use occurs

). The effect of the binding would be tested or demonstrated by powerless dragging (6).

A B

C

(1)

(2)

(3) (4)

(5)

urther modalities arise if the same dragging acts in another articulation this will occur if one recon-

ructs the problem slightly by removing k. Then one might ask: When is the angle at C acute, when is it

btuse? This may again be explored via wandering dragging. But then the problem solver becomes per-

aps interested in figuring out when the angle is exactly a right angle. To that end he will use guidedragging. In order to explore the locus of all these boundary points he might try to move C so that the

ngle remains constant i.e. lieu muet dragging. It may be helpful to mark points on this locus to make it

sible. Utilizing the trace like in the figure would be another possible articulation of the same dragging

ct.

o far, the dragged point was also the object, whose movement was to be observed. By relaxing this con-

tion, one arrives at some additional dragging modalities. These would readily apply in the following

xample 2 Given a triangle ABC, when lies its orthocenter H inside ABC, if you vary the position of C?

bserve that these examples only exhibit heuristic use of DGS in Boeros phases I and II. A heuristic us-ge in phase III occurs in the following

xample 3 If the problem solver of example 1 is to formulate and prove a theorem, he may arrive at


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qual to the angles at A and B (Decomposing and Recombining orInvariance Principle). Using theRe-

ated Solved Problem that the measures of the interior angles of a Euclidean triangle always add up to 180

egrees the proof can be finished.

he complexity of Heuristics

ur schemes of heuristic reconstruction and instrumentation serve several purposes one is to secure that

n the analysis of tasks no possibilities to reconstruct and instrument are overlooked. This will be helpfulith demanding tasks. Also, our schemes help to clarify what makes heuristic work difficult. For instancecan be readily seen that application of Polyas scheme is not a straightforward affair by trying to recon-

ruct the following example forWorking Backwards:

rimary task (Hlzl 1996, 184) Consider a triangle ABC inscribed in a circle k with A varying on k

how that the locus M of the midpoint M of AB is a circle.

B

C

A

M

s reconstruction may again start with discarding the hypothesis to give students the opportunity to go

hrough Boeros phases I and II. The task is then suitable to illustrate the use of several of Polyas original

uestions - phase III can even be framed as an inner dialogue in the spirit of Polyas teacher-student-

ialogues:What is the unknown! A circle. What is the condition? Every circle is determined by center and radius.

ry to think of a familiar problem having the same or a similar unknown. Many locus problems have a

rcle as their solution. Could you imagine a more accessible related problem? We look for a circlesenter. The given circle has also a center.Introduce suitable notation. It is the circumcenter O of ABC.ould you restate the related problem? If ABC is inscribed in a circle k, the center of k coincides with the

rcumcenter O. Could you change the unknown so that the new unknown and the new data are nearer to

ach other? To that end, one may consider that OMB is also a triangle that is inscribed in a circle. Could

ou restate the problem? One could ask: how is the unknown circle determined by OMB? Could you re-ate it still differently? Of course, the unknown circle is the circumcircle of OMB. But that does not seem

o be directly helpful. Could you imagine a more accessible related problem? Did you use all the

ata? Indeed, OMB is inscribed in a special way: OB is a diameter of the unknown circle. This means

hat one can apply Thales'theorem!

ow the approach to Work Backwards starts to look promising: Using the converse of Thales'theorem,

ne can prove the desired conclusion if one is able to prove that angle OMB is always a right angle. This

oal can be achieved in the same vein: using the hypothesis as premise, one derives that OMB is the mir-


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or image of OMB with respect to the reflection axis OM. Thus AOB is isosceles. By inverting this line of

ought one readily arrives at the desired hypothesis, thereby completing phase IV successfully.

O

B

C

A

M

owever, one sees that the solution path jumps backwards and forward all over Polyas scheme. It is

nlikely that actual learning trajectory proceed in this way but it will be fruitful to compare them after

e solution has been found. Looking back in this way may serve to illustrate the value of heuristic and

ontribute to remember some of the questions more easily in future problems.


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ow can DGS be utilized in solving this problem? If one sticks to the solution path above, it can only

ay a minor role: e.g. step 21. is supported if one observes the invariance of the right angle at M by ex-

oratively dragging C along k. This occurs indeed, see Hlzl (1996, 185). But furthermore the dynamic

isualizations entails a conceptual reframing of the problem situation: the solvers view it dynamically

nstead of statically namely it occurs to them that M arises from A by a dilatation with center B. Now it

ecomes pretty clear that the locus must be a circle dilatations preserve distances!

his empirical observation illustrates how far-reaching the influence of DGS on a proof process can be.

CASE STUDY

inally, we want to elucidate the virtues of an elaborate dragging practice in an advanced problem solvingtuation. Some years ago we were able to observe the work of Deinhardt, a heuristically talented primary

acher student. For her final examination thesis we instructed her to delve into an example already recon-

ructed by Hlzl:

rimary taskShow that the locus H of the orthocentre H of a triangle ABC inscribed in a circle k with

varying on k is the reflection of k by AB.

raditionally, one proves this via transformation geometry, but runs into trouble when in phase IV tryingo justify rigorously the visually obvious existence of the reflection. Since the needed background is fal-

ng off in schools anyway, this should not be reckoned as a well suited entre problem - but even worse:

lzl (2001) reports empirical evidence that for many students even the hypothesis itself does not seem to

e worth noting: since C varies on a circle, they deemed it plausible that the locus is also a circle. In order

o elucidate its peculiarity, Hlzl has proposed to untie B from k and then explore the situation thereby

rescribing the strategy of generalization! Dragging B in this way unfolds a variety of curves that may

imulate analytical work to explain the peculiarity of the special case.

n those grounds, Deinhardt was instructed to investigate:

How can the manifold of occurring curves be ordered and described?

. Can something interesting be said about this manifold?he task was set quite broadly to allow for heuristic work. Deinhardt solved it so splendidly that the re-

ults were published (Deinhardt 2000)! We can only sketch them:


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Deinhardt succeeds to construct a universal familyHM,B of orthocentre loci that realizes all possible

onfigurations up to similarity. With coordinates it can be described as follows: A is fixed to the origin, B

linked to the positive x-axis. C then varies on a circle of radius 1 with centre M on the unit circle. Whenarying M and B, one successively covers all possible configurations. Deinhardt orders them according to

eirgestalt.

M

BA

C

ntroducing coordinates allows Deinhardt to utilize the heuristic strategy Transformation: Via the Carte-

an correspondence the geometric construction ofH can betranslated to an algebraic description: Ele-

entary analytic geometry yields a parameterization ofH. By eliminating the parameter one gets an

quation forH that depends on M and B.

Deinhardt chooses to inspect the occurring singularities more closely. The exploration ofHM,B leads

er to the observation that H can have at most one singular point: either a node or a cusp. As can been

een above, nodes occur quite frequently, whereas cusps are more rare. When do they occur?

or a concise description of the cusp case, Deinhardt unties B from the x-axis. By lieu muet dragging, she

dentifies some positions of B such thatHM,B has a cusp. It looks as if these points lie on a circle:


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ut further investigation disproves this conjecture: near A the locus of cusp points itself seemingly has a

usp!

y meticulously exploring the situation geometrically and analyzing the results algebraically, Deinhardt

ucceeds in uncovering a surprising result: It turns out that the locus of cusp points B (i.e. position of B

uch thatHM,B has a cusp) is a cardioid!

einhardt is able to again utilize Transformation to confirm here findings by analytic geometry. She thusves a stunning example for a double instrumental genesis: By applying advanced heuristic strategies

nd instrumenting them with refined DGS utilizations schemes (lieu muet dragging, binding and untying

oints) Deinhardt is able to achieve far more than one would have ever reckoned


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GS can here be seen to advance the solution process by alleviating the switch between ascending and

escending control movements sensu Arzarello as well as the change of enactive, iconic and symbolicepresentation modes sensu Bruner instead of a detailed analysis we must refrain ourselves to a sche-

matic diagram:

ONCLUSIONhe example of Deinhardt shows how heuristic problem solving is carried forward by skilful utilization of

arious dragging modalities. It is not meant to suggest that most or many students will reach this level of

mastery by our heuristic training. But we will investigate the long-term impact of integrating EiWos intoaily classroom practice, using standard as well as non-standard problems as benchmarks. And we do ex-

ect that this training will significantly increase the heuristic literacy of our students.

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ck, Th. & Henn, H.-W. (eds.)Zeichnung Figur Zugfigur, Hildesheim: Franzbecker


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oero, P. (1999). Argumentation and mathematical proof: A complex, productive, unavoidable relation-

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rockmann-Behnsen, D. (2011, in press): A long-term educational treatment using dynamic geometry

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lschenbroich, H.-J. (1997). Tod des Beweisens oder Wiederauferstehung? In H. Hischer (ed.) Computer

nd Geometrie. Neue Chancen fr den Geometrieunterricht?. Hildesheim: Franzbecker

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reudenthal H. (1973)Mathematics as an educational task, Dordrecht : Reidelawlick, Th. (2002): On Dynamic Geometry Software in the Regular Classroom.Zentralblatt fr Didak-

k der Mathematik34(3), 85-92

awlick, Th. (2003): Utilizing DGS as heuristic tool in introductory calculus courses. Proceedings of

CTME 2. Beirut.

awlick, Th. (2005): Connecting Arguments to Actions Dynamic Geometry as Means for the Attain-

ent of higher van Hiele Levels.Zentralblatt fr Didaktik der Mathematik, 37(5).

ayes, J.(1966): Memory, goals and problem solving. In: B.Kleinmuntz (ed.): Problem Solving:Research,

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efendehl-Hebeker, L.& Hlzl, R.( 2000): Zerlegbare Dreiecke im problemorientierten Unterricht. In:

aptist, P. (ed.):Mathematikunterricht im Wandel, Bamberg:Buchner.

ealy, L., Holzl, R., Hoyles, C., & Noss, R. (1994): Messing up.Micromath, 10(1), 14-16.lzl, R. (1996): How does 'dragging' affect the learning of geometry?Int. J. Computers for Math.

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livero, F. (1999): Cabri-Gomtre as a mediator in the process of transition to proofs in open geometric

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ducational Computing Research, 10(2), 173-197.efouad B. (2000). Etude de genses instrumentales lies lutilisation dune calculatrice symbolique en

asse de premire S. Thse de doctorat. Universit Paris 7.e Villiers, M. (1997): The Role of Proof in Investigative, Computer-based Geometry: Some personal

eflections. In Schattschneider, D. & King,J. (eds.) Geometry Turned On! Washington: MAA.

e Villiers, M. (2003): Rethinking Proof with Sketchpad 4. Emeryville: Key Curriculum Press.

e Villiers, M. (2007): Some pitfalls of dynamic geometry software, Teaching & Learning Mathematics,

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rijvers P. (2000). Students encountering obstacles using CAS. The International Journal of Computers

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lzl, R. (1994).Im Zugmodus der Cabri-Geometrie. Interaktionsstudien und Analysen zum Mathematik-rnen mit dem Computer. Weinheim: Deutscher Studien Verlag

uin D. & Trouche L. (1999). The complex process of converting tools into mathematical instruments :he case of calculators. The International Journal of Computers for Mathematical Learning, 3(3), 195-227.

aborde, C. (1998). Visual phenomena in the teaching/learning of geometry in a computer-based envi-onment. In C. Mammana & V. Villani (eds.), Perspectives on the teaching of geometry for the 21stcen-

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ctivity of proving.Educational Studies in Mathematics, 44 (1-3), 151-161.aborde, C.(2001): Integration of Technology in the Design of Geometry tasks with Cabri-Geometry.Int.

Computers for Math. Learn., 6(3), 283-317.


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aborde, C., Kynigos, C., Hollebrands, K., & Strer, R. (2006). Teaching and learning geometry with

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livero, F. (2002). Proving within dynamic geometry environments, Ph. D. Thesis, Graduate School of

ducation, Bristololya, G. (1954).Mathematics and Plausible Reasoning. Induction and Analogy in Mathematics. Vol 1.rinceton: Princeton University Presschumann, H. (1991): Interactive generalizing of geometric configurations,Int. J. Computers for Math.earn., 22(6), 953 - 963

trer, R. (2001). Chancen und Probleme des Zugmodus. In: Elschenbroich H.-J., Gawlick, Th. &enn, H.-W. (eds.)Zeichnung Figur Zugfigur, Hildesheim: Franzbecker.

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trategies.Journal for Research in Mathematics Education, 21(5), 411 415

oeplitz, O. (1963). The Calculus, a genetic approach. Chicago: University Press.

EY TERMS & DEFINITIONS

euristic literacy: Students ability to apply heuristics.

euristic reconstruction: Method to design teaching units to foster heuristic literacy utilizing Polyas

roblem solving scheme.

euristic instrumentation: Method to integrate DGS in the acquisition of heuristic skills by conscious use

f dragging modalities

ragging modalities: Dragging acts that are performed in various articulations.

iWos: Electronic interactive Worksheets backing the heuristic instrumentation of problems.

click drag think prefinal

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